Optimal. Leaf size=65 \[ -\frac {(5 a x+i) \sqrt {a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.20, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5085, 5082, 81} \[ -\frac {(5 a x+i) \sqrt {a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 81
Rule 5082
Rule 5085
Rubi steps
\begin {align*} \int \frac {e^{5 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{5 i \tan ^{-1}(a x)} x^2}{\left (1+a^2 x^2\right )^{27/2}} \, dx}{c^{13} \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \frac {x^2}{(1-i a x)^{16} (1+i a x)^{11}} \, dx}{c^{13} \sqrt {c+a^2 c x^2}}\\ &=-\frac {(i+5 a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 63, normalized size = 0.97 \[ \frac {(1-5 i a x) \sqrt {a^2 x^2+1}}{120 a^3 c^{13} (a x-i)^{10} (a x+i)^{15} \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 497, normalized size = 7.65 \[ \frac {{\left (i \, a^{22} x^{25} - 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} - 50 i \, a^{18} x^{21} - 126 \, a^{17} x^{20} - 280 i \, a^{16} x^{19} - 160 \, a^{15} x^{18} - 765 i \, a^{14} x^{17} + 105 \, a^{13} x^{16} - 1248 i \, a^{12} x^{15} + 720 \, a^{11} x^{14} - 1260 i \, a^{10} x^{13} + 1260 \, a^{9} x^{12} - 720 i \, a^{8} x^{11} + 1248 \, a^{7} x^{10} - 105 i \, a^{6} x^{9} + 765 \, a^{5} x^{8} + 160 i \, a^{4} x^{7} + 280 \, a^{3} x^{6} + 126 i \, a^{2} x^{5} + 50 \, a x^{4} + 40 i \, x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{120 \, a^{27} c^{14} x^{27} + 600 i \, a^{26} c^{14} x^{26} + 120 \, a^{25} c^{14} x^{25} + 5400 i \, a^{24} c^{14} x^{24} - 6000 \, a^{23} c^{14} x^{23} + 19920 i \, a^{22} c^{14} x^{22} - 39600 \, a^{21} c^{14} x^{21} + 34320 i \, a^{20} c^{14} x^{20} - 125400 \, a^{19} c^{14} x^{19} + 6600 i \, a^{18} c^{14} x^{18} - 241560 \, a^{17} c^{14} x^{17} - 99000 i \, a^{16} c^{14} x^{16} - 300960 \, a^{15} c^{14} x^{15} - 237600 i \, a^{14} c^{14} x^{14} - 237600 \, a^{13} c^{14} x^{13} - 300960 i \, a^{12} c^{14} x^{12} - 99000 \, a^{11} c^{14} x^{11} - 241560 i \, a^{10} c^{14} x^{10} + 6600 \, a^{9} c^{14} x^{9} - 125400 i \, a^{8} c^{14} x^{8} + 34320 \, a^{7} c^{14} x^{7} - 39600 i \, a^{6} c^{14} x^{6} + 19920 \, a^{5} c^{14} x^{5} - 6000 i \, a^{4} c^{14} x^{4} + 5400 \, a^{3} c^{14} x^{3} + 120 i \, a^{2} c^{14} x^{2} + 600 \, a c^{14} x + 120 i \, c^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a x + 1\right )}^{5} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {27}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 58, normalized size = 0.89 \[ \frac {\left (-a x +i\right ) \left (a x +i\right ) \left (5 a x +i\right ) \left (i a x +1\right )^{5}}{120 a^{3} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 46, normalized size = 0.71 \[ -\frac {c\,{\left (a\,x-\mathrm {i}\right )}^5\,\left (5\,a\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{120\,a^3\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{29/2}\,\sqrt {a^2\,x^2+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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